Problem 1. Engineering Math
Evaluate
$\iint\limits_S y dS$
Where $S$ is the portion of the cylinder $x^2+y^2=k^2$ that lies between $z=a$ and $z=b$.
Solution: The parameterization of this cylinder
$\vec{r}(\theta, z)=(k\cos\theta, k\sin\theta, z)$ where $a\leq z\leq b$ and $0\leq\theta \leq 2\pi$
$\vec{r}_\theta(\theta, z)=\dfrac{d\vec{r}}{d\theta}=(-k\sin\theta, k\cos\theta, 0)$
$\vec{r}_z(\theta, z)=\dfrac{d\vec{r}}{dz}=(0, 0, 1)$
The cross product is
$\vec{r}_\theta\times \vec{r}_z=\left|{\begin{bmatrix}
\vec{i}&\vec{j}&\vec{k}\\ -k\sin\theta & k\cos\theta & 0\\ 0 & 0 & 1
\end{bmatrix}}\right|=(-k\cos\theta, -k\sin\theta, 0)$
The norm of the cross product is $\|{\vec{r}_\theta\times \vec{r}_z}\|=k$. Then
$\iint\limits_S y dS=\iint\limits_D k\sin\theta\cdot kdA=\int\limits_0^{2\pi}\int\limits_a^b k^2\sin\theta \quad dzd\theta=0.$
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Problem 2. Calculus.
Quadratic Approximation
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Problem 3. Linear Algebra.
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